Local coefficient system

A local coefficient system assigns an Abelian group or module to each point, with compatible transport isomorphisms along paths depending only on homotopy relative endpoints. Equivalently it is a functor from the fundamental groupoid. On a connected manifold it is described by a representation of the fundamental group after fixing one fiber.

Its fiber at $x$ is $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$. Transport changes its generator by the sign of local orientation along the path. For a connected nonorientable manifold, its degree-zero cohomology is zero, because no nonzero integer is invariant under sign reversal, whereas its degree-zero homology is $\mathbb{Z}/2$, the coinvariants under that reversal.